Homogeneous Mixtures and Molecular Cross-sections

Macroscopic Cross-section of a Homogeneous Mixture

Let’s consider a homogeneous mixture of two nuclides $X$ and $Y$. The atomic densities of each nuclide are $N_X$ and $N_Y$ $\text{atom/cm}^3$ respectively, and the cross-sections for a specific reaction between neutrons and these nuclei are $\sigma_X$ and $\sigma_Y$ respectively.

Then, the probability of a neutron colliding with nuclei $X$ and $Y$ per unit length is $\Sigma_X=N_X\sigma_X$ and $\Sigma_Y=N_Y\sigma_Y$ respectively (refer to Macroscopic Cross-section). The total probability of a neutron reacting with these two types of nuclei per unit length is as follows:

\[\Sigma = \Sigma_X + \Sigma_Y = N_X\sigma_X + N_Y\sigma_Y \tag{1}\]

Equivalent Cross-section of a Molecule

If the nuclei we examined above exist in molecular form, we can define the equivalent cross-section of that molecule by dividing the macroscopic cross-section of the mixture obtained from equation (1) by the number of molecules per unit volume.

If there are $N$ molecules of $X_mY_n$ per unit volume, then $N_X=mN$, $N_Y=nN$, and from equation (1), we can calculate the cross-section of this molecule as follows:

\[\sigma = \frac{\Sigma}{N}=m\sigma_X + n\sigma_Y \tag{2}\]

Equations (1) and (2) are valid under the assumption that nuclei $X$ and $Y$ react independently with neutrons, so they cannot be applied to elastic scattering by molecules and solids. The scattering cross-sections of molecules and solids at low neutron energies must be determined through experiments.